WASHINGTON — John F. Kelly, the secretary of the Department of Homeland Security, eased a key part of President Trump’s immigration order on Sunday, saying that people from the affected countries who hold green cards will not be prevented from returning to the United States.

“In applying the provisions of the president’s executive order, I hereby deem the entry of lawful permanent residents to be in the national interest,” Mr. Kelly said in a statement. “Accordingly, absent the receipt of significant derogatory information indicating a serious threat to public safety and welfare, lawful permanent resident status will be a dispositive factor in our case-by-case determinations.”